Thread: differential analysis

Find an example of a differentiable function

f : R → R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).

Suppose that

f : R → R has derivatives of all orders. Prove that the same

is true of F(x) := exp (f(x)).

not sure on either of these two questions..

thanks in advance

Originally Posted by James0502

Find an example of a differentiable function

f : R → R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).

Suppose that

f : R → R has derivatives of all orders. Prove that the same

is true of F(x) := exp (f(x)).

not sure on either of these two questions..

thanks in advance

For the first one, the following link might be helpful Weierstrass function - Wikipedia, the free encyclopedia, and for the second one, you may refer to http://www.mathhelpforum.com/math-he...-new-post.html
About the first one I really wonder if there is an explicit answer, as I remember such a discussion was made previously .

Originally Posted by James0502

Find an example of a differentiable function

f : R → R such that f'(0) = 1 > 0 but f is not monotonic increasing on any interval (0,delta).

Suppose that

f : R → R has derivatives of all orders. Prove that the same

is true of F(x) := exp (f(x)).

not sure on either of these two questions..

thanks in advance

For the first question, you need to be differentiable everywhere, but must be discontinuous at 0 (otherwise, it would be positive near 0 and would be increasing).

A quite famous example of a differentiable function which is not continuously differentiable at 0 is the following: with . However, , so we should change it a little... You can prove that the function with is an answer to your problem. The "2" is probably optional but may simplify the proof.

You have to prove that is differentiable on (straightforward), continuous at 0, differentiable at 0 (look at ), that and that has not a constant sign on any neighbourhood of 0.